Question: Multiply the following complex numbers: $({2-2i}) \cdot ({-4-5i})$
Explanation: Complex numbers are multiplied like any two binomials. First use the distributive property: $ ({2-2i}) \cdot ({-4-5i}) = $ $ ({2} \cdot {-4}) + ({2} \cdot {-5}i) + ({-2}i \cdot {-4}) + ({-2}i \cdot {-5}i) $ Then simplify the terms: $ (-8) + (-10i) + (8i) + (10 \cdot i^2) $ Imaginary unit multiples can be grouped together. $ -8 + (-10 + 8)i + 10i^2 $ After we plug in $i^2 = -1$ , the result becomes $ -8 + (-10 + 8)i - 10 $ The result is simplified: $ (-8 - 10) + (-2i) = -18-2i $